D-dimensional Model Research Articles

A simple formula for the critical temperature of D-dimensional Ising models

Abstract We propose a simple equation for the critical temperature (CT) of D-dimensional Ising models
(on squared or not squared lattices). When 2 < D < ∞, these models have no exact solution
and even the simpler problem of locating the CT analytically remains a formidable task. The
mean-field as well as the belief-propagation methods provide approximations that work only for
large D, remaining quite unsatisfactory for D ≤ 4. Our simple equation represents a sort of “finite-
dimensional” generalization of such approaches. It provides a reasonable approximation for D = 2
and works pretty well for D ≥ 3 where, in general, outperforms other purely analytical methods.
The equation is derived by using duality relations and heuristic arguments.

Generalized approach for the perturbative dynamical braneworld in D dimensions

In this paper, we propose an approach to derive the brane cosmology in the D-dimensional braneworld model. We generalize the “bulk-based” approach by treating the 4-brane as a small perturbation to the D-dimensional spherically symmetric spacetime. The linear corrections from a static 4-brane to the metric are derived from the linearized perturbation equations, while the nonlinear corrections are found by a parameterization of the perturbed metric solution. We use a time-dependent generalization to give the nonlinearly perturbed metric solution for the dynamical braneworld model, and analyze the stability of the model under the motion of the 4-brane. Through the fine tuning, we can recover the Friedmann equations for the universe with and without an effective cosmological constant. More importantly, the de Sitter expansion of the universe can be reproduced.

New models of d-dimensional black holes without inner horizon and with an integrable singularity

Theoretically, it has been proposed that objects traveling radially along regular black holes (RBHs) would not be destroyed because of finite tidal forces and the absence of a singularity. However, the matter source allows the creation of an inner horizon linked to an unstable de Sitter core due to mass inflation instability. This inner horizon also gives rise to the appearance of a remnant, inhibiting complete evaporation. We introduce here a d-dimensional black hole model with Localized Sources of Matter (LSM), characterized by the absence of an inner horizon and featuring a central integrable singularity instead of an unstable de Sitter core. In our model, any object tracing a radial and timelike world-line would not be crushed by the singularity. This is attributed to finite tidal forces, the extendability of radial geodesics, and the weak nature of the singularity. Our LSM model enables the potential complete evaporation down to r h = 0 without forming a remnant. In higher dimensions, complete evaporation occurs through a phase transition, which could occur at Planck scales and be speculatively driven by the Generalized Uncertainty Principle (GUP). Unlike RBHs, our model satisfies the energy conditions. We demonstrate a linear correction to the conventional area law of entropy, distinct from the RBH's correction. Additionally, we investigate the stability of the solutions through the speed of sound.

Exotic swarming dynamics of high-dimensional swarmalators.

Swarmalators are oscillators that can swarm as well as sync via a dynamic balance between their spatial proximity and phase similarity. Swarmalator models employed so far in the literature comprise only one-dimensional phase variables to represent the intrinsic dynamics of the natural collectives. Nevertheless, the latter can indeed be represented more realistically by high-dimensional phase variables. For instance, the alignment of velocity vectors in a school of fish or a flock of birds can be more realistically set up in three-dimensional space, while the alignment of opinion formation in population dynamics could be multidimensional, in general. We present a generalized D-dimensional swarmalator model, which more accurately captures self-organizing behaviors of a plethora of real-world collectives by self-adaptation of high-dimensional spatial and phase variables. For a more sensible visualization and interpretation of the results, we restrict our simulations to three-dimensional spatial and phase variables. Our model provides a framework for modeling complicated processes such as flocking, schooling of fish, cell sorting during embryonic development, residential segregation, and opinion dynamics in social groups. We demonstrate its versatility by capturing the maneuvers of a school of fish, qualitatively and quantitatively, by a suitable extension of the original model to incorporate appropriate features besides a gallery of its intrinsic self-organizations for various interactions. We expect the proposed high-dimensional swarmalator model to be potentially useful in describing swarming systems and programmable and reconfigurable collectives in a wide range of disciplines, including the physics of active matter, developmental biology, sociology, and engineering.

Global regularity for Oldroyd-B model with only stress tensor dissipation

In this paper, we consider the d-dimensional ( d ⩾ 2) Oldroyd-B model with only dissipation in the equation of stress tensor, and establish a small data global well-posedness result in critical L p framework. Particularly, we give a positive answer to the problem proposed recently by Wu-Zhao (J. Differ. Equ. 316 (2022)) involving the upper bound for the time integral of the lower frequency piece of the stress tensor, and show that it is indeed independent of the time. Moreover, we improve the results in (J. Math. Fluid Mech. 24 (2022)) by relaxing the space dimension d = 2 , 3 to any d ⩾ 2.

Convergence of densities of spatial averages of the parabolic Anderson model driven by colored noise

In this paper, we present a rate of convergence in the uniform norm for the densities of spatial averages of the solution to the d-dimensional parabolic Anderson model driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. The proof is based on the combination of Malliavin calculus techniques and the Stein's method for normal approximations.

Application of the variational autoencoder to detect the critical points of the anisotropic Ising model.

We generalize the previous study on the application of variational autoencoders to the two-dimensional Ising model to a system with anisotropy. Due to the self-duality property of the system, the critical points can be located exactly for the entire range of anisotropic coupling. This presents an excellent test bed for the validity of using a variational autoencoder to characterize an anisotropic classical model. We reproduce the phase diagram for a wide range of anisotropic couplings and temperatures via a variational autoencoder without the explicit construction of an order parameter. Considering that the partition function of (d+1)-dimensional anisotropic models can be mapped to that of the d-dimensional quantum spin models, the present study provides numerical evidence that a variational autoencoder can be applied to analyze quantum systems via the quantum Monte Carlo method.

Solvable Dynamics of Coupled High-Dimensional Generalized Limit-Cycle Oscillators.

We introduce a new model consisting of globally coupled high-dimensional generalized limit-cycle oscillators, which explicitly incorporates the role of amplitude dynamics of individual units in the collective dynamics. In the limit of weak coupling, our model reduces to the D-dimensional Kuramoto phase model, akin to a similar classic construction of the well-known Kuramoto phase model from weakly coupled two-dimensional limit-cycle oscillators. For the practically important case of D=3, the incoherence of the model is rigorously proved to be stable for negative coupling (K<0) but unstable for positive coupling (K>0); the locked states are shown to exist if K>0; in particular, the onset of amplitude death is theoretically predicted. For D≥2, the discrete and continuous spectra for both locked states and amplitude death are governed by two general formulas. Our proposed D-dimensional model is physically more reasonable, because it is no longer constrained by fixed amplitude dynamics, which puts the recent studies of the D-dimensional Kuramoto phase model on a stronger footing by providing a more general framework for D-dimensional limit-cycle oscillators.

High-Degree Collisional Moments of Inelastic Maxwell Mixtures—Application to the Homogeneous Cooling and Uniform Shear Flow States

The Boltzmann equation for d-dimensional inelastic Maxwell models is considered to determine the collisional moments of the second, third and fourth degree in a granular binary mixture. These collisional moments are exactly evaluated in terms of the velocity moments of the distribution function of each species when diffusion is absent (mass flux of each species vanishes). The corresponding associated eigenvalues as well as cross coefficients are obtained as functions of the coefficients of normal restitution and the parameters of the mixture (masses, diameters and composition). The results are applied to the analysis of the time evolution of the moments (scaled with a thermal speed) in two different nonequilibrium situations: the homogeneous cooling state (HCS) and the uniform (or simple) shear flow (USF) state. In the case of the HCS, in contrast to what happens for simple granular gases, it is demonstrated that the third and fourth degree moments could diverge in time for given values of the parameters of the system. An exhaustive study on the influence of the parameter space of the mixture on the time behavior of these moments is carried out. Then, the time evolution of the second- and third-degree velocity moments in the USF is studied in the tracer limit (namely, when the concentration of one of the species is negligible). As expected, while the second-degree moments are always convergent, the third-degree moments of the tracer species can be also divergent in the long time limit.

On the classical critical behavior of the specific heat capacity of a model of structural phase transitions with a long-range interaction

The critical specific heat capacity c of a d-dimensional model describing structural phase transitions in an anharmonic crystal with a long-range interaction (decreasing at large distances r as r −d−σ , 0 < σ ≤ 2) is studied near the classical critical point Tc . At temperatures T > Tc and for dimensions σ < d < 2σ (σ and 2σ are the lower and the upper critical dimensions, respectively) the critical specific heat capacity is obtained in the form c ≈ 1 − Dεαs , where D > 0 and αs < 0 depend only on the ratio d/σ, and ε = T/Tc −1 is a measure of the deviation from the critical point. For three fixed values of the ratio d/σ the dependence c ≈ c(ε) is graphically presented. It is shown that at all temperatures T ≤ Tc the specific heat capacity retains its maximum value, c max = 1. The critical exponent αs , obtained here, coincides with that of the known mean spherical model, while c max is different for the two models.

On Fluxbrane Polynomials for Generalized Melvin-like Solutions Associated with Rank 5 Lie Algebras

We consider generalized Melvin-like solutions corresponding to Lie algebras of rank 5 (A5, B5, C5, D5). The solutions take place in a D-dimensional gravitational model with five Abelian two-forms and five scalar fields. They are governed by five moduli functions Hs(z) (s=1,...,5) of squared radial coordinates z=ρ2, which obey five differential master equations. The moduli functions are polynomials of powers (n1,n2,n3,n4,n5)=(5,8,9,8,5),(10,18,24,28,15),(9,16,21,24,25),(8,14,18,10,10) for Lie algebras A5, B5, C5, D5, respectively. The asymptotic behavior for the polynomials at large distances is governed by some integer-valued 5×5 matrix ν connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in A5 and D5 cases) with the matrix representing a generator of the Z2-group of symmetry of the Dynkin diagram. The symmetry and duality identities for polynomials are obtained, as well as asymptotic relations for solutions at large distances.

Uniform bounds for ruin probability in multidimensional risk model

In this paper we consider some generalisations of the classical d-dimensional Brownian risk model. This contribution derives some non-asymptotic bounds for simultaneous ruin probabilities of interest. In addition, we obtain non-asymptotic bounds also for the case of general trend functions and convolutions of our original risk model.

A parallel computing method for the higher order tensor renormalization group

In this paper, we propose a parallel computing method for the Higher Order Tensor Renormalization Group (HOTRG) applied to a d-dimensional (d≥2) simple lattice model. Sequential computation of the HOTRG requires O(χ4d−1) computational cost, where χ is bond dimension, in a step to contract indices of tensors. When we simply distribute elements of a local tensor to each process in parallel computing of the HOTRG, frequent communication between processes occurs. The simplest way to avoid such communication is to hold all the tensor elements in each process, however, it requires O(χ2d) memory space. In the presented method, placement of a local tensor element to more than one process is accepted and sufficient local tensor elements are distributed to each process to avoid communication between processes during considering computation step. For the bottleneck part of computational cost, such distribution is achieved by distributing elements of two local tensors to χ2 processes according to one of the indices of each local tensor which are not contracted during considering computation. In the case of d≥3, computational cost in each process is reduced to O(χ4d−3) and memory space requirement in each process is kept to be O(χ2d−1).

Coulomb gas and sine-Gordon model in arbitrary dimension

The sine-Gordon (SG), i.e. periodic scalar field theory is known to play an important role in d=2 dimensions. A paradigmatic example is the topological phase transition of the vortex dynamics in superfluid films and layered superconductors which are described by SG type models. Periodic scalar potentials find applications in d=4 dimensions, too. Higgs, inflaton and axion physics are examples where scalar fields naturally appear, thus, the SG model can be used instead of the usual polynomial one. The SG quantum field theory can be mapped onto the neutral Coulomb-gas (CG) in arbitrary dimension and the renormalization group (RG) study of the d-dimensional CG model was obtained in the dilute gas approximation. It signals a single phase for d>2, however, it was shown recently, that a suitable generalization of the SG model can posses a topological phase transitions in d=4 dimensions. Our goals in this work are (i) to map out the phase structure of the (original) SG and the equivalent neutral CG models by the functional RG method in arbitrary dimension, (ii) to compare the 3-dimensional SG and isotropic XY spin models and show that they belong to different universality classes, (iii) to study the consequences of the findings on higgs, inflaton, axion models and on the topological phase transition in higher dimensions.

Effects of topological characteristics on rhythmic states of the D-dimensional Kuramoto model in complex networks.

Synchronization is a ubiquitous phenomenon in engineering and natural ecosystems. While the dynamics of synchronization modeled by the Kuramoto model are commonly studied in two dimensions and the state of dynamic units is characterized by a scalar angle variable, we studied the Kuramoto model generalized to D dimensions in the framework of a complex network and utilized the local synchronous order parameter between the agent and its neighbors as the controllable variable to adjust the coupling strength. Here, we reported that average connectivity of networks affects the time-dependent, rhythmic, cyclic state. Importantly, we found that the level of heterogeneity of networks governs the rhythmic state in the transition process. The analytical treatment for observed scenarios in a D-dimensional Kuramoto model at D=3 was provided. These results offered a platform for a better understanding of time-dependent swarming and flocking dynamics in nature.

Chemical Subdiffusivity of Critical 2D Percolation

We show that random walk on the incipient infinite cluster (IIC) of two-dimensional critical percolation is subdiffusive in the chemical distance (i.e., in the intrinsic graph metric). Kesten (Probab Theory Related Fields 73(3): 369–394 1986) famously showed that this is true for the Euclidean distance, but it is known that the chemical distance is typically asymptotically larger. More generally, we show that subdiffusivity in the chemical distance holds for stationary random graphs of polynomial volume growth, as long as there is a multiscale way of covering the graph so that “deep patches” have “thin backbones.” Our estimates are quantitative and give explicit bounds in terms of the one and two-arm exponents \(\eta _2> \eta _1 > 0\): For d-dimensional models, the mean chemical displacement after T steps of random walk scales asymptotically slower than \(T^{1/\beta }\), whenever $$\begin{aligned} \beta < 2 + \frac{\eta _2-\eta _1}{d-\eta _1}\,. \end{aligned}$$Using the conjectured values of \(\eta _2 = \eta _1 + 1/4\) and \(\eta _1 = 5/48\) for 2D lattices, the latter quantity is \(2+12/91\).

Sharp decay estimates for Oldroyd-B model with only fractional stress tensor diffusion

Precise large-time behavior of physical quantities plays a crucial role in understanding many physical phenomena. For partial differential equation (PDE) models with full dissipation, powerful methods such as the Fourier-splitting technique have been developed. However, these methods may not be applied to PDE systems with only partial dissipation. This paper offers new ideas on how to obtain precise large-time decay estimates on a partially dissipated system. We examine the d-dimensional incompressible Oldroyd-B model without velocity dissipation and with only fractional diffusive stress. The discovery here is that the coupling and interaction of the velocity and the non-Newtonian stress actually enhances the regularity and the stability of the system. Without the stress, the Sobolev norms of the velocity could grow rather rapidly in time, let alone decay at explicit rates. Making use of the interaction, we deduce a system of damped wave equations obeyed by the velocity and the Leray projection of the divergence of the stress. By constructing a suitable Lyapunov functional, we are able to control the growth in the derivatives and extract explicit decay rates. The optimal decay rates are established by representing the wave equations in an integral form and applying a bootstrapping argument.

Ott–Antonsen ansatz for the D-dimensional Kuramoto model: A constructive approach

Kuramoto’s original model describes the dynamics and synchronization behavior of a set of interacting oscillators represented by their phases. The system can also be pictured as a set of particles moving on a circle in two dimensions, which allows a direct generalization to particles moving on the surface of higher dimensional spheres. One of the key features of the 2D system is the presence of a continuous phase transition to synchronization as the coupling intensity increases. Ott and Antonsen proposed an ansatz for the distribution of oscillators that allowed them to describe the dynamics of the order parameter with a single differential equation. A similar ansatz was later proposed for the D-dimensional model by using the same functional form of the 2D ansatz and adjusting its parameters. In this article, we develop a constructive method to find the ansatz, similarly to the procedure used in 2D. The method is based on our previous work for the 3D Kuramoto model where the ansatz was constructed using the spherical harmonics decomposition of the distribution function. In the case of motion in a D-dimensional sphere, the ansatz is based on the hyperspherical harmonics decomposition. Our result differs from the previously proposed ansatz and provides a simpler and more direct connection between the order parameter and the ansatz.

Pandemic-type failures in multivariate Brownian risk models

Modelling of multiple simultaneous failures in insurance, finance and other areas of applied probability is important especially from the point of view of pandemic-type events. A benchmark limiting model for the analysis of multiple failures is the classical d-dimensional Brownian risk model (Brm), see Delsing et al. (Methodol. Comput. Appl. Probab. 22(3), 927–948 2020). From both theoretical and practical point of view, of interest is the calculation of the probability of multiple simultaneous failures in a given time horizon. The main findings of this contribution concern the approximation of the probability that at least k out of d components of Brm fail simultaneously. We derive both sharp bounds and asymptotic approximations of the probability of interest for the finite and the infinite time horizon. Our results extend previous findings of Dȩbicki et al. (J. Appl. Probab. 57(2), 597–612 2020) and Dȩbicki et al. (Stoch. Proc. Appl. 128(12), 4171–4206 2018).

Global large solution for the tropical climate model with diffusion

We study the d-dimensional (d=2,3) tropical climate model with only the dissipation of the first baroclinic model of the velocity −ηΔv. By choosing a class of special initial data (u0,v0,𝜃0) whose Hs(ℝd) norm can be arbitrarily large, we obtain the global smooth solution of d-dimensional tropical climate model.